U-statistics have a well developed theory, which provides information on their asymptotic properties [1]}, [2]}, [3]}, [4]}. In the generic case \(\sqrt{n} U_n\)  tends to a Gaussian, but degenerate cases are scaled as \(n^{r/2}U_n\)  and tend to other limit laws, with the rank \(r\)  defined as follows.
